NOTE. In mechanics, motion in a straight line is represented by a segment, the direction of the line indicating the direction of the motion, and the length of the segment representing the magnitude of the motion.

10. Three men start from St. Louis: the first going 8 miles eastward; the second going 12 miles southward; the third going 16 miles southwesterly. Represent these journeys by line segments.

N

Represent the position of St. Louis by the point 0. Through O draw two perpendicular lines WE and NS as shown. The directions E, N, W, S are represented by the sense of the half lines OE, ON, OW, OS, respectively. Choose a convenient scale, say, 1 inch = 4 miles. On OE lay off OA = 2 inches, on OS lay W off OB = 3 inches; draw OD, making an angle of 45° with OS, and lay off OC = 4 inches. Then the segments OA, OB, OC will represent the journeys, the arrowheads indicating the directions of the motions.

B
IS

11. An attempt is made to row a boat at the rate of 4 miles an hour directly across a stream flowing at the rate of 3 miles an hour. Represent the motion of the boat.

A

We assume the motion of the boat to be the resultant of two independent component motions, one due to the propulsion of the oars and one due to the force of the current. That is, in a given time the resultant of the two component motions acting simultaneously is assumed to be the same as if they acted consecutively, or as if the boat first floated down stream for a given unit of time without propulsion of the oars and was then rowed across through still water for an equal period of time. Therefore, the construction is as follows:

BY

C

Draw OA 4 units in length. This will represent the component motion in a given time due to the propulsion of the oars. Draw OB OA, making OB 3 units in length. This will represent the component motion due to the stream. Complete the parallelogram OBCA, and draw the diagonal OC. OC will represent the motion of the boat in the given unit of time.

NOTE. When two component uniform motions acting along straight lines have different directions, the process of finding the resultant involves what is known as THE principle oF THE PARALLELOGRAM OF MOTIONS, Viz.: If the segments which represent the components are made adjacent sides of a parallelogram, the diagonal drawn from the angle included by these sides will represent the resultant in both magnitude and direction.

12. A flag is drawn steadily downward 60 feet from the masthead of a moving ship. During the same time, the ship moves forward 24 feet. Represent the direction and length of the actual path of the flag.

13. A sailor climbs a mast at the rate of 3 feet a second. The ship is sailing at the rate of 12 feet a second. Represent the path through which he actually moves during 20 seconds.

14. A railway train is moving northeastward at the rate of 30 miles an hour. (a) Represent the rate at which it is moving eastward; (b) represent the rate at which it is moving northward.

15. A train is moving with a speed of 3 miles an hour. A brakeman on the top of the train runs toward the engine with a speed of 10 feet a second. What is his resultant speed? What would be his resultant speed if he were to run with the same speed toward the rear of the train?

16. If the boatman in Ex. 11 wished to go straight across the stream in spite of the current, represent the direction which he must row.

17. A sledge party is traveling on the ice toward the north pole at the rate of one mile an hour. The ice is drifting southeastward at the rate of 22 yards a minute. Represent the actual path of the party.

18. A football receives, simultaneously, two horizontal blows, — one from the west, having a force of 18 lb., and one from the southeast, having a force of 12 lb. Represent the direction of its motion.

By

By the second law of motion (see works on physics) the motion imparted by each blow is in the direction of the impressed

C

force and is proportional to it.

Hence, construct

a parallelogram having adjacent sides 18 units and 12 units with included angle 135°. Then the A diagonal drawn from the included angle will represent the direction of the football.

NOTE. Since motion takes place in the direction of the impressed force and is proportional to it, the diagonal OC also represents a single force which is equivalent to the given forces OA and OB. Hence, to find the resultant of two concurrent forces acting in different directions, proceed as in finding the resultant of two component motions.

19. A football receives, simultaneously, three horizontal blows, one from the north, having a force of 10 lb. ; one from the east, having a force of 15 lb. ; and one from the southeast, having a force of 25 lb. Represent the resultant of the three forces, and the direction of the motion of the football.

20. A woman pushes on the handle of a carpet sweeper with a force of 5 lb. If the handle makes an angle of 60° with the floor, represent the effective force which acts horizontally to move the sweeper; also the force

which is exerted perpendicularly on the sweeper. What is the effective force in pounds tending to move the sweeper?

21. A team of horses pulls a log by a chain inclined at an angle of 30° with the horizontal. If the force exerted on the chain is 500 lb., represent the effective horizontal force, and the vertical force which tends to lift the log. What is the force in pounds which tends to lift the log?

22. Two equal forces acting at an angle of 120° have a resultant of 50 lb. What is the magnitude of each force?

23. A 200 lb. weight is supported by two strings, each making an angle of 30° with the horizontal. What is the tension of each string?

24. The difference of the perpendiculars from any point on the base produced of an isosceles triangle to the equal sides of the triangle, or the equal sides produced, is equal to the altitude upon one of the equal sides.

G

Draw CHIDE. CG HE. It remains to prove B that DF

=

DH

E

H

D

25. The sum of the perpendiculars from any point within an equilateral triangle to the three sides is equal to the altitude.

Draw MN through 0 || BC. OD + OF AG.

(Why?)

CHAPTER V

THE CIRCLE AND THE REGULAR POLYGON. LOCI OF POINTS

P

A

282. Extension of the notion of angle. In Chapter I (§ 61) we defined an angle as the figure formed by two half lines which issue from the same point. The term "angle" is used also, however, in a somewhat different sense, which may be made clear as follows: When a wheel rotates on its axis, the amount of rotation is conveniently measured by the angle through which it turns. Thus, we say 66 a wheel rotates through an angle of 30° or 60°" and the meaning to be attached to the phrase is evident. Again, in ten minutes, the minute hand of a clock "rotates through an angle of 60°," i.e. † of a complete revolution. In order to make this use of an angle as a measure of rotation applicable in all cases, however, we must recognize as angles the result of a rotation through more than 180°.

B

FIG. 124.

Thus, if in the figure (Fig. 124) we suppose the half line OP to have rotated from its initial position OA in the direction of the arrowhead to its terminal position OB, the angle which measures this rotation is an angle of 230°. (With our previous notion of angle, where we considered only the figure of the two half lines, we should have considered this an angle 130°.)

283. Our new notion of angle may be formulated as follows: Imagine a half line OP to rotate about the (fixed) point O from any initial position, as OA, to some terminal position, as OB. The rotating line is then said to generate an angle, and the size of the angle is the amount of rotation that has taken place. Such an angle is called an angle of rotation, when it is necessary to distinguish it from the older more restricted notion.

110

An angle of rotation may have any size whatever. Thus, if the half line OP makes one complete revolution, it has generated an angle equal to 360° or 4 R. If it continues to rotate in the same sense through another right angle, it will have generated an angle of 450°; and so on.

284. Equal angles of rotation. Two angles of rotation are equal only if they represent the same amount of rotation. Thus, an angle of 90° and one of 450° are not equal, even though the two figures representing the angles are congruent.

B

B

285. In particular, if two angles of rotation ABC and A'B'C' are equal and less than two right angles, the angles ABC and A'B'C' are congruent in the older sense of the word angle. Therefore, in discussions where the size of angles is limited to angles that are less than two right angles, the term "angle may be used in either sense without confusion. 286. Oblique and reflex angles. An angle which is less than two right angles is called an oblique angle; an angle which is greater than two right angles but less than four right angles is called a reflex angle. An angle which is equal to two right angles is called a straight angle.

A

Reflex Angle

FIG. 125.

Oblique Angle

Two half lines issuing from a point O (see Fig. 124) form two angles, one oblique and one reflex, unless the two half lines are in the same straight line. To distinguish between these two angles use is usually made of arcs of circles having their center at O and terminating in the sides of the angle.

ANGLES FORMED BY STRAIGHT LINES WHICH MEET A CIRCLE

287. An inscribed angle. The (oblique) angle formed by two chords which meet on a circle is said to be inscribed in the circle. The angle is said to be subtended by the arc which lies within the angle and is cut off by the sides.

An angle is said to be inscribed in an arc of a circle if its vertex lies on the arc and its sides pass through the extremities of the arc. Thus, the angle ABC is inscribed in the arc ABC and is subtended by the arc ADC.

B

C

D

FIG. 126.